# Utilizing SVD and VMD for Denoising Non-Stationary Signals of Roller Bearings

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## Abstract

**:**

## 1. Introduction

- (1)
- SVD and difference spectrum (DS) of singular value are proposed to determine the number of intrinsic mode functions (IMFs) of VMD;
- (2)
- The effective order of SVD and the kurtosis of VMD are used to select the IMFs of VMD, and non-stationary signals are denoised by reconstructing the selected components of VMD;
- (3)
- The effectiveness and performance of the proposed method is verified using simulation bearing data and real experimental bearing data. These results are compared with those from SVD-DS.

## 2. Theoretical Basis

#### 2.1. Singular Value Decomposition (SVD)

#### 2.2. Difference Spectrum of Singular Values

#### 2.3. Variational Mode Decomposition (VMD)

_{k}has a central frequency and a finite bandwidth. To evaluate the bandwidth of each mode, the corresponding constrained variational problem is given as follows:

## 3. The Proposed SVD-VMD Methodology

## 4. Simulative Case Study

_{out}and RMSE by the abovementioned methods for the noisy simulated non-stationary signal with different $SN{R}_{in}$ from −20 dB to 5 dB is displayed in Table 3.

## 5. Experimental Case Study

^{5}points. The experimental rotating frequency is about 30 Hz. The vibration signal of the rolling bearing with the inner ring failure is shown in Figure 9. The calculated defect frequency is 5.4152 times the shaft rotational speed (Hz). Since the shaft rotational speed is 1797 rpm (corresponding to the rotational frequency fr = 29.2 Hz), the inner ring fault frequency is 162.19 Hz.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Time-domain waveform and frequency spectrum of impulsive signal. (

**a**) Waveform of impulsive signal; (

**b**) Spectrum of impulsive signal.

**Figure 3.**Time-domain waveform and frequency spectrum of simulated signal. (

**a**) Waveform of simulated signal (

**b**) spectrum of simulated signal.

**Figure 5.**Decomposition result of VMD for the simulated signal. (

**a**) Time domain decomposition result; (

**b**) Frequency domain decomposition result.

**Figure 8.**Roller bearing test rig [36].

**Figure 9.**Time-domain waveform and frequency spectrum of vibration signal with inner ring failure. (

**a**) Time-domain waveform of vibration signal with inner ring failure; (

**b**) Frequency spectrum of vibration signal with inner ring failure; (

**c**) Frequency spectrum limited in 0.9 KHz of vibration signal with inner ring failure.

**Figure 10.**Time-domain waveform and frequency spectrum of denoised signal using SVD-VMD. (

**a**) Time-domain waveform of denoised signal using SVD-VMD; (

**b**) Frequency spectrum of denoised signal using SVD-VMD; (

**c**) Frequency spectrum limited in 0.9 KHz of denoised signal using SVD-VMD.

IMF | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Kurtosis value | 29.06 | 31.18 | 24.52 | 71.65 |

SVD-DS | SVD-VMD | |
---|---|---|

SNR_{out} (dB) | −7.16 | −0.65 |

RMSE | 0.49 | 0.23 |

SNR_{in} | SNR_{out} of SVD-DS | SNR_{out} of SVD-VMD | RMSE of SVD-DS | RMSE of SVD-VMD |
---|---|---|---|---|

−20 | −18.09 | −7.66 | 1.72 | 0.52 |

−10 | −17.57 | 1.57 | 1.62 | 0.18 |

−5 | −17.31 | 4.19 | 1.57 | 0.13 |

−1 | −15.44 | 4.24 | 1.27 | 0.13 |

5 | −14.52 | 3.41 | 1.14 | 0.14 |

IMF | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Kurtosis value | 34.05 | 31.67 | 32.41 | 34.75 | 20.76 | 50.10 | 26.04 | 30.88 |

SVD-DS | SVD-VMD | |
---|---|---|

SNR_{out} (dB) | −6.58 | 1.60 |

RMSE | 1.10 | 0.43 |

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**MDPI and ACS Style**

Wang, Q.; Wang, L.; Yu, H.; Wang, D.; Nandi, A.K.
Utilizing SVD and VMD for Denoising Non-Stationary Signals of Roller Bearings. *Sensors* **2022**, *22*, 195.
https://doi.org/10.3390/s22010195

**AMA Style**

Wang Q, Wang L, Yu H, Wang D, Nandi AK.
Utilizing SVD and VMD for Denoising Non-Stationary Signals of Roller Bearings. *Sensors*. 2022; 22(1):195.
https://doi.org/10.3390/s22010195

**Chicago/Turabian Style**

Wang, Qinghua, Lijuan Wang, Hongtao Yu, Dong Wang, and Asoke K. Nandi.
2022. "Utilizing SVD and VMD for Denoising Non-Stationary Signals of Roller Bearings" *Sensors* 22, no. 1: 195.
https://doi.org/10.3390/s22010195